Homology index braids in infinite-dimensional Conley index theory

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1 CADERNOS DE MATEMÁTICA 05, October (2004) ARTIGO NÚMERO SMA#208 Homology index braids in infinite-dimensional Conley index theory Maria C. Carbinatto * Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, São Carlos SP, BRASIL mdccarbi@icmc.usp.br Krzysztof P. Rybakowski Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, Rostock, GERMANY krzysztof.rybakowskimathematik.uni-rostock.de We extend the notion of a categorial Conley-Morse index, as defined in [20], to the case based on a more general concept of an index pair introduced in [12]. We also establish a naturality result of the long exact sequence of attractorrepeller pairs with respect to the choice of index triples. In particular, these results immediately give a complete and rigorous existence result for homology index braids in infinite dimensional Conley index theory. Finally, we describe some general regular and singular continuation results for homology index braids obtained in our recent papers [6] and [7]. October, 2004 ICMC-USP 1. INTRODUCTION The concept of the categorial Morse index for flows on locally compact spaces is a refinement of Conley index. It was developed by Conley [8] and his students (mainly Kurland [14]). Roughly speaking, the categorial Morse index (or Conley-Morse index) I(S) of a compact isolated invariant set S (relative to a given flow) is a connected simple system and a subcategory of the homotopy category of pointed spaces with objects (N 1 /N 2, [N 2 ]) where (N 1, N 2 ) is an index pair in some compact isolating neighborhood N of S. The morphisms of I(S) are inclusion or flow induced. Later Franzosa [9], [11], [10] used a somewhat more general concept of an index pair and an ensuing categorial Conley-Morse index, more suitable for applications to Morse-decompositions and homology index braids. The Conley index theory and the categorial Conley-Morse index were extended by Rybakowski [19], [20] to semiflows on (not necessarily locally compact) metric spaces. The isolating neighborhoods in that theory are required to satisfy an admissibility condition, making the theory applicable to various classes of evolution equations. The concept of index pairs in this extended theory is analogous to that used in [8] and [14]. * The research of the first author was partially supported by the grant CNPq Brazil /

2 290 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI Parts of Franzosa s theory of Morse-decompositions and homology index braids were extended by Franzosa and Mischaikow [12] to the setting of [19] and [20]. These authors use a definition of index pairs which is the analogue of Franzosa s definition in the locally compact case. Motivated by [12] we define in the present paper a categorial Conley-Morse index C(S) whose objects are generated by index pairs in the sense of [12] (rather than index pairs as defined in [19]). We also establish existence of C(S) (Propositions 4.18and 4.19) and prove that certain types of inclusion induced morphisms lie in C(S) (Propositions 4.21 and 4.22). These results are not only of interest in themselves but they are also needed for a precise definition of long exact sequences of attractor-repeller pairs in the non-locally compact case considered here. Simplifying slightly the approach of Kurland [16] we also define the category of connected simple systems in a given category K. Moreover, for a given connected simple system C in K and a functor Φ from K to a module category, we define the image module ˆΦ(C) (cf Section 3). All this allows us, in Section 5, to define the long exact homology sequence of an attractorrepeller pair (A, A ) in S, associated with a given index triple (N 1, N 2, N 3 ) for (S, A, A ). In Theorem 5.26 we prove that this sequence is independent of the choice of (N 1, N 2, N 3 ). These results also resolve some technical issues which remained open in the derivation of the homology index braid as outlined in [12] (the hints given on pp of [12] are insufficient for that). In particular, we can now proceed exactly as in [9] and [12] to obtain a precise definition of the homology index braid for a given (partially ordered) Morse-decomposition. This is done in Section 6, in which we also discuss morphisms from one homology index pair to another. In particular, we define inclusion induced morphisms between homology index braids and show that, under a certain nesting property, these morphism are isomorphisms. In Section 7, which is based on our recent paper [6], we consider a sequence π n, n N 0, of local semiflows on X and a sequence (π n, S n, (M p,n ) p P ), n N 0, of Morse-decompositions such that (π n, S n, (M p,n ) p P ) regularly converges to (π 0, S 0, (M p,0 ) p P ). We state the nested index filtration theorem (Theorem 7.35), which immediately implies a general (regular) continuation result for homology index braids and Morse-decompositions (Theorems 7.36 and 7.38). We apply this result to Galerkin approximations of semilinear parabolic equations. Finally, in Section 8, based on our recent work [7], we state a nested index filtration theorem in the context of singular perturbation problems (Theorem 8.48), which implies a general singular continuation result for homology index braids and connection matrices (Theorem 8.49). We apply this result to reaction-diffusion equations on thin domains. 2. PRELIMINARIES The purpose of this section is to recall a few concepts from Conley index theory and to establish some preliminary results needed later in this paper. We assume the reader s familiarity with the (infinite-dimensional) Conley index theory, as expounded in the papers [19] and [20] (or the book [21]), and with the papers [9], [11] and [12]. Sob a supervisão da CPq/ICMC

3 HOMOLOGY INDEX BRAIDS 291 Let X be a topological space. Choose an arbitrary, but fixed point p / X. Let A, Y be subspaces of X. Suppose first that Y A. Define an equivalence relation on Y by letting x y if and only if x = y or x, y Y A. We denote by Y/A the quotient space of Y modulo this equivalence relation. We write [A] to denote the equivalence class of any member x of Y A. Set-theoretically, [A] = Y A. We endow Y/A with the quotient topology. Now let Y A =. We endow the set X := X {p} with the sum topology, i.e. U is open in X {p} if and only if U X is open in X. Setting Y := Y {p}, A := {p} we define Y /A and [A ] as above and set Y/A := Y /A and [A] := [A ]. Note that [A] = {p} this time. With our choice p / X the following simple result holds. Proposition 2.1. If A Y X then the pair (Y, A) is uniquely determined by the pointed space (Y/A, [A]). Proof. If A, then A = [A] while Y is the union of all equivalence classes of the relation, i.e. Y = Y/A. If A =, then Y = { y X {y} Y/A }. Remark 2.2. If A, Y are subspaces of a topological space X, we will often denote the pointed space (Y/A, [A]) simply by Y/A. This should not lead to confusion. For the rest of this paper, unless otherwise specified, X is a metric space, π is a local semiflow on X and all (the relevant) concepts are defined relative to π. Suppose that Y is a subset of X. By Inv + π (Y ), resp. Inv π (Y ), resp. Inv π (Y ) we denote the largest positively invariant, resp. negatively invariant, resp. invariant subset of Y. Moreover, let the function ρ Y = ρ Y,π : Y R { } be given by ρ Y (x) := sup{ t 0 xπt is defined and xπ[0, t] Y }. (2.1) It is clear that if Y, Y X and x Y Y, then ρ Y Y (x) = min(ρ Y (x), ρ Y (x)). (2.2) Y is called π-admissible if Y is closed and whenever (x n ) n and (t n ) n are such that t n and x n π [0, t n ] Y for all n N, then the sequence (x n πt n ) n has a convergent subsequence. We say that π does not explode in Y if whenever x X and xπt Y as long as xπt is defined, then xπt is defined for all t [0, [. Y is called strongly π-admissible if Y is π-admissible and π does not explode in Y. Let N and Y be subsets of X. The set Y is called N-positively invariant if whenever x Y, t 0 are such that xπ [0, t] N, then xπ [0, t] Y. Let N, Y 1 and Y 2 be subsets of X. The set Y 2 is called an exit ramp for N within Y 1 if whenever x Y 1 and xπt N for some t [0, [, then there exists a t 0 [0, t ] such that xπ [0, t 0 ] N and xπt 0 Y 2.

4 292 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI If Y 1 and Y 2 are subsets of X then Y 2 is called an exit ramp for Y 1 if Y 2 is an exit ramp for N within Y 1, where N = Y 1. Definition 2.3. Let B X be a closed set and x B. The point x is called a strict egress (respectively strict ingress, respectively bounce-off ) point of B, if for every solution σ : [ δ 1, δ 2 ] X of π through x, with δ 1 0 and δ 2 > 0, the following properties hold: 1. There exists an ε 2 ]0, δ 2 [ such that σ(t) B (respectively σ(t) Int X (B), respectively σ(t) B), for t ]0, ε 2 ]. 2. If δ 1 > 0, then there exists an ε 1 ]0, δ 1 [ such that σ(t) Int X (B) (respectively σ(t) B, respectively σ(t) B), for t [ ε 1, 0[. The set of all strict egress (respectively strict ingress, respectively bounce-off) points of B is denoted by B e (respectively B i, respectively B b ). Moreover, we call B := B e B b the exit set of B and B + := B i B b the entrance set of B. B is called an isolating block, if B = B e B i B b and B is closed. If B is also an isolating neighborhood of an invariant set S, then we say that B is an isolating block for S. If B is an isolating block then (B, B ) is an example of an index pair in B. generally, we have the following definition. Definition 2.4. (relative to π) if: More Let N be closed in X. A pair (N 1, N 2 ) is called an index pair in N 1. N 1 and N 2 are closed and N-positively invariant subsets of N; 2. N 2 is an exit ramp for N within N 1 ; 3. Inv π (N) is closed and Inv π (N) Int X (N 1 \ N 2 ). Definition 2.5. A pair (N 1, N 2 ) is called a Franzosa-Mischaikow-index pair (or FMindex pair) for (π, S) if: 1. N 1 and N 2 are closed subsets of X with N 2 N 1 and N 2 is N 1 -positively invariant; 2. N 2 is an exit ramp for N 1 ; 3. S is closed, S Int X (N 1 \ N 2 ) and S is the largest invariant set in Cl X (N 1 \ N 2 ); Proposition 2.6. [cf. [12]] Let (N 1, N 2 ) be a pair of closed subsets of X with N 2 N If S is an isolated invariant set, N 1 is an isolating neighborhood of S and (N 1, N 2 ) is an index pair in N 1, then (N 1, N 2 ) is an FM-index pair for (π, S). 2. If (N 1, N 2 ) is an FM-index pair for (π, S) and N is an isolating neighborhood of S with N 1 \ N 2 N, then N 1 N is an isolating neighborhood of S and (N 1 N, N 2 N) is an index pair in N 1 N. Sob a supervisão da CPq/ICMC

5 HOMOLOGY INDEX BRAIDS 293 Proposition 2.7. Let N 1, N 2 and N be closed subsets of X with N 1 \ N 2 N. Then the inclusion induced map j : (N 1 N)/(N 2 N) N 1 /N 2 is an isomorphism in the category of pointed spaces. Proof. Proposition I.6.2 in [21] implies that j is a continuous map. Moreover, there is an inclusion induced map (in the sense of Definition I.6.1 in [21]) k : N 1 /N 2 (N 1 N)/(N 2 N) which is also continuous (by Proposition I.6.2 in [21]). We need to show that k is the inverse of j. Let z (N 1 N)/(N 2 N). If z = [x], where x (N 1 N)\(N 2 N), then j(z) = [x] N 1 /N 2 and so k(j(z)) = [x] (N 1 N)/(N 2 N). Otherwise, z = [N 2 N] and j(z) = [N 2 ]. Thus, k(j(z)) = [N 2 N], since k and j are base-point preserving maps. Let z N 1 /N 2. If z = [x], where x N 1 \ N 2, then k(z) = [x] (N 1 N)/(N 2 N) and x (N 1 N) \ (N 2 N). Therefore, j(k(z)) = [x] N 1 /N 2. Otherwise, z = [N 2 ] and k(z) = [N 2 N] and so j(k(z)) = [N 2 ]. Definition 2.8. Let S be a compact invariant set and (A, A ) be an attractor-repeller pair in S, relative to π. A pair (B 1, B 2 ) is called a block pair (for (π, S, A, A )) if B 1 is an isolating block for A, B 2 is an isolating block for A, B := B 1 B 2 is an isolating block for S and B 1 B 2 B1 B+ 2. If (B 1, B 2 ) is a block pair then (B, B 2 B, B ) is an example of an FM-index triple: Definition 2.9. Let S be a compact invariant set and (A, A ) be an attractor-repeller pair in S relative to π. A triple (N 1, N 2, N 3 ) with N 3 N 2 N 1 is called an FM-index triple (for (π, S, A, A )) if (N 1, N 3 ) is an FM-index pair for (π, S) and (N 2, N 3 ) is an FM-index pair for (π, A). Proposition (cf. [12]) Let (N 1, N 2, N 3 ) be an FM-index triple for (π, S, A, A ). Then (N 1, N 2 ) is an FM-index pair for (π, A ). For the rest of this paper we fix a (commutative) ring Γ and a Γ-module G. Given a chain complex C, we denote by H q (C), q Z, the homology of C with coefficients in G. Recall (cf. [11]) that a sequence C 1 i C 2 p C 3

6 294 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI of chain maps is called weakly exact if ker i = 0, p i = 0 and the map H q (ρ) from H q (C 2 / im i) to H q (C 3 ) is an isomorphism for each q Z. Here, the map ρ: C 2 / im i C 3 is the (uniquely determined) chain map with ρ Q = p, where Q: C 2 C 2 / im i is the quotient map. Given a weakly exact sequence C 1 i C 2 p C 3 and given q Z, define the map q : H q (C 3 ) H q 1 (C 1 ) by q := q H q (ρ) 1, where q : H q (C 2 / im i) H q 1 (C 1 ) is the connecting homomorphism in the long exact sequence induced by the short exact sequence 0 i C 1 C Q 2 C 2 / im i 0. Using elementary homology theory we obtain the following result. Proposition (cf. [11]) Given a weakly exact sequence C 1 i C 2 p C 3 the corresponding homology sequence H q (C 1 ) H q(i) H q (C 2 ) H q(p) H q (C 3 ) q H q 1 (C 1 ) is exact. Moreover, given a commutative diagram C 1 i C 2 p C 3 f 1 C 1 of chain maps with weakly exact rows, the induced long homology ladder ĩ C2 f 2 p C3 f 3 H q (C 1 ) Hq(i) H q (C 2 ) Hq(p) H q (C 3 ) q H q 1 (C 1 ) H q (f 1 ) H q (f 2 ) H q (f 3 ) H q ( C 1 ) H q ( C 2 ) H q ( C 3 ) Hq (ĩ) Hq ( p) H q 1 (f 1 ) q H q 1 ( C 1 ) is commutative. Sob a supervisão da CPq/ICMC

7 HOMOLOGY INDEX BRAIDS 295 If Y is a topological space, then (Y ) denotes the singular chain complex (see [23]). If (Y, B) is a topological pair, we define As usual, we set Thus, C(Y/B) = C(Y/B, {[B]}) := (Y/B)/ ({[B]}). H q (Y/B) := H q (C(Y/B)), q Z. for each q Z, H q (Y/B) is the q-th singular homology group of the pair (Y/B, {[B]}), with coefficients in G. (2.3) Proposition (cf. [11] and [12]) Suppose that (N 1, N 2, N 3 ) is an FM-index triple for (π, S, A, A ) with Cl X (N 1 \ N 3 ) strongly π-admissible. Then the inclusion induced sequence N 2 /N 3 i N 1 /N 3 p N 1 /N 2 (2.4) of pointed spaces induces a weakly exact sequence C(N 2 /N 3 ) i C(N 1 /N 3 ) p C(N 1 /N 2 ) of chain maps. Propositions 2.11 and 2.12 thus imply the following result. Proposition Suppose that (N 1, N 2, N 3 ) is an FM-index triple for (π, S, A, A ) with Cl X (N 1 \ N 3 ) strongly π-admissible. Then the long sequence H q (N 2 /N 3 ) H q(i) H q (N 1 /N 3 ) H q(p) H q (N 1 /N 2 ) q H q 1 (N 2 /N 3 ) (2.5) induced by (2.4) is exact. There is a similar result for Alexander-Spanier cohomology [23]. More precisely, let H q, q Z, denote the q-th Alexander-Spanier cohomology functor with values in G. If Y and B are closed in X and B Y, then the strong excision property of Alexander-Spanier cohomology implies that the quotient map Q = Q Y,B : (Y, B) (Y/B, {[B]})

8 296 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI induces a module isomorphism H q (Q): H q (Y/B, {[B]}) H q (Y, B), q Z. Again we set H q (Y/B) := H q (Y/B, {[B]}) for short. In other words, for q Z, H q (Y/B) is the q-th Alexander-Spanier cohomology group of the pair (Y/B, {[B]}), with coefficients in G. (2.6) Therefore, given a triple (N 1, N 2, N 3 ) of closed sets in X with N 1 N 2 N 3 we can define, for each q Z, the map by q : H q+1 (N 2 /N 3 ) H q (N 1 /N 2 ) q = H q (Q N1,N 2 ) 1 q H q+1 (Q N2,N 3 ), where q : H q+1 (N 2, N 3 ) H q (N 1, N 2 ) is the connecting homomorphism of the exact cohomology sequence for the triple (N 1, N 2, N 3 ). From the cohomology sequence of space triples we thus obtain the following result. Proposition Let (N 1, N 2, N 3 ) be a triple of closed sets in X with N 1 N 2 N 3. Then inclusion induced sequence N 2 /N 3 i N 1 /N 3 p N 1 /N 2 induces a long exact cohomology sequence H q (i) H q (p) H q (N 2 /N 3 ) H q (N 1 /N 3 ) H q (N 1 /N 2 ) H q+1 (N 2 /N 3 ). q 3. CATEGORIES OF CONNECTED SIMPLE SYSTEMS In this section, simplifying a little the approach by Kurland [16], we will define categories of connected simple subsystems of a given category. We will also define images of connected simple systems under functors with values in a category of modules. These notions are required for a precise development of the categorial Conley-Morse index and the long exact (co)homology sequence of an attractor-repeller pair. LetK be a fixed category. The letters C, C and C will denote subcategories of K which are connected simple systems. Sob a supervisão da CPq/ICMC

9 HOMOLOGY INDEX BRAIDS 297 Given objects A, B in C and objects A, B in C and α Mor K (A, A ), β Mor K (B, B ) we say that α is related to β in K relative to (C, C ) (and we write αϱ C,C β or just αϱβ) if and only if the following diagram commutes (in K): f A α A f B β B. Here, f (resp. f ) are the unique elements of Mor C (A, B) (resp. Mor C (A, B )). Since f and f are isomorphisms (in K), it follows that αϱβ implies βϱα. Moreover, the diagram commutes, so αϱα. If the diagrams Id A A α A A α A Id A f A α A f and g B β B B β B C γ C commute, then so does the diagram A α A g g f C γ C. g f Thus αϱβ and βϱγ imply that αϱγ. It follows that ϱ = ϱ C,C on the set is an equivalence relation Ω(C, C ) := { Mor K (A, A ) A Obj(C) and A Obj(C ) }. (3.1) Given α Ω(C, C ), let [α] = [α] ϱc,c be the equivalence class of α. We define a category [K] whose objects are all the subcategories of K which are connected simple systems. Given objects C and C in K, let Mor [K] (C, C ) be the set of all ζ for which there is an α Ω(C, C ) with ζ = [α]. (In order to make the morphism sets mutually disjoint, as is required in the definition of a category, one should more precisely consider ordered triples (ζ, C, C ) rather than just ζ to be morphisms from C to C. We shall not bother, however.)

10 298 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI Given ζ Mor [K] (C, C ) and ζ Mor [K] (C, C ) let α: A A and α : C A be such that ζ = [α] and ζ = [α ]. Let f be the unique element of Mor C (A, C ) and define ζ ζ := [α f α]. We claim that this definition is independent of the choice of α and α. In fact, if β : B B and β : D B are such that ζ = [β] and ζ = [β ] and g : B D is the unique element of Mor C (B, D ), then the following diagram commutes: A α A f C α A B β B g D β B. Here, the vertical arrows are the unique morphisms in the respective connected simple systems. It follows that (α f α)ϱ C,C (β g β) and so [α f α] = [β g β] which proves our claim. Thus the composition ζ ζ is well-defined and is clearly associative. Indeed, the consideration of sequences of the form A α A f C α A f C α A implies that (α f α ) f α = α f (α f α). Thus, if ζ = [α], ζ = [α ] and ζ = [α ], we have (ζ ζ ) ζ = [α f α ] [α] = [(α f α ) f α] and ζ (ζ ζ) = [α ] [α f α] = [α f (α f α)]. Hence, (ζ ζ ) ζ = ζ (ζ ζ). Moreover, the commutativity of the diagram A Id A A f B IdB B, f where f is the unique element of Mor C (A, B), shows that Id A ϱ Id B and so [Id A ] = [Id B ] for any two objects in C. We set Id C := [Id A ], where A is any object in C. Clearly whenever ζ Mor [K] (C, C ), then there are objects A and A in C and C respectively, such that ζ = [α], where α Mor K (A, A ). Thus Id A α = α and α Id A = α so Id C ζ = ζ and ζ Id C = ζ. It follows that Id C is an identity for the composition in [K] and so [K] is, indeed, a category, which we term the category of connected simple systems in K. If C, C are objects in [K] and α Ω(C, C ) (with Ω(C, C ) being defined in (3.1)), then ζ := [α] ϱc,c is called the morphism in [K] induced by α, relative to (C, C ). Sob a supervisão da CPq/ICMC

11 HOMOLOGY INDEX BRAIDS 299 Remark The present definition of the category [K], while conceptually (hopefully!) simpler, is equivalent to the definition of the category CSS(K) given in [16] in the sense that [K] and CSS(K) are isomorphic categories. Now, suppose Φ is a covariant functor from K to the category Mod(Γ) of modules over the (commutative) ring Γ. Let C be an object of [K]. Let S = S C be the disjoint union of all Φ(A), where A is an arbitrary object of C. Thus, formally we have S = S C := A Obj(C) On S = S C define a relation R = R C as follows: Φ(A) {A}. (x, A)R(y, B) if and only if y = Φ(f)x, where f is the unique morphism in C from A to B. Clearly, R is an equivalence relation on S. Let S/R be the set of equivalence classes of R and Q = Q C : S S/R given by Q ((x, A)) = [(x, A)] R for (x, A) S be the canonical quotient map. In the sequel we write Φ(C) := S/R. For each A Obj(C), the map Q A = Q C,A : Φ(A) Φ(C) given by Q A (x) = Q ((x, A)) for x Φ(A) is easily seen to be bijective. Moreover, if (x, A)R(y, B) and ( x, A)R(ỹ, B), then (x + A x, A)R(y + B ỹ, B) and (λ A x, A)R(λ B y, B) for every λ Γ. Here, for every C Obj(C), + C (resp. C) is the addition (resp. scalar multiplication) in the Γ-module Φ(C).Therefore, there is a unique addition + = + C and scalar multiplication = C in Φ(C) such that for every A Obj(C), the map Q A is a Γ-module isomorphism. The Γ-module Φ(C) is called the image module of C under Φ. Now let C and C be objects of [K] and A Obj(C), A Obj(C ) be arbitrary. If F is a morphism in Mod(Γ) from Φ(A) to Φ(A ), then define the map F : Φ(C) Φ(C ) by Then F is a Γ-module homomorphism. Moreover, F := Q C,A F Q C,A 1. Proposition Suppose A, B Obj(C), A, B Obj(C ). If the diagram Φ(f) Φ(A) Φ(B) F G Φ(A ) Φ(f ) Φ(B ),

12 300 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI commutes, then F = G, where f (resp. f ) is the unique morphism in C (resp. C ) from A to B (resp. from A to B ). Proof. Let η Φ(C) be arbitrary. Then there exist an x Φ(A) and a y Φ(B) such that η = Q ((x, A)) = Q ((y, B)). It follows that y = Φ(f)x. Now and F (η) = (Q A F Q A 1 )(η) = Q A F x G (η) = (Q B G Q B 1 )(η) = Q B (Gy) = Q B G(Φ(f)x) = Q B (Φ(f )F x). Notice that Q B (Φ(f )F x) = Q ((Φ(f )F x, B )) = Q ((F x, A )) = Q A (F x). This implies that F (η) = G (η). The proposition is proved. The following result is obvious. Proposition Let C, C and C be objects of [K] and A Obj(C), A Obj(C ), A Obj(C ) be arbitrary. Let F be a morphism in Mod(Γ) from Φ(A) to Φ(A ) and F be a morphism in Mod(Γ) from Φ(A ) to Φ(A ) then F F = F F. If (F, F ) is exact, i.e., ker F = im F, then so is ( F, F ). We call the assignment F F the -operation (associated with Φ). 4. THE CATEGORIAL CONLEY-MORSE INDEX In this section we will extend the notion of the categorial Morse index from [20] in the sense that index pairs (and quasi-index pairs) will be replaced by FM-index pairs. Let K be the homotopy category of pointed spaces. If π is a local semiflow defined in a metric space X and S is an isolated π-invariant set admitting a strongly π-admissible isolating neighborhood, then we define a subcategory C(S) = C(π, S) as follows. The objects of C(S) are the pointed spaces (E, p) = (N 1 /N 2, [N 2 ]), where (N 1, N 2 ) is an FMindex pair for (π, S) and Cl X (N 1 \ N 2 ) is strongly π-admissible. Given two objects (E, p) and (Ẽ, p) in C(S), Proposition 2.1 implies that there are unique FM-index pairs (N 1, N 2 ) and (Ñ1, Ñ2) for (π, S) with Cl X (N 1 \ N 2 ) and Cl X (Ñ1 \ Ñ2) strongly π-admissible such that (E, p) = (N 1 /N 2, [N 2 ]) and (Ẽ, p) = (Ñ1/Ñ2, [Ñ2]). Let N and Ñ be arbitrary strongly π-admissible isolating neighborhoods of S with N 1 \ N 2 N and Ñ1 \ Ñ2 Ñ (e.g. we may take N = Cl X(N 1 \ N 2 ) and Ñ = Cl X(Ñ1 \ Ñ2)). Then Proposition 2.6 implies that (N 1 N, N 2 N) and (Ñ1 Ñ, Ñ2 Ñ) are index pairs in N 1 N and Ñ1 Ñ, respectively. Therefore, there is a unique morphism τ : N 1/N 2 Ñ1/Ñ2 in K making the following diagram commutative in K. Sob a supervisão da CPq/ICMC

13 HOMOLOGY INDEX BRAIDS 301 (N 1 N)/(N 2 N) β (Ñ1 Ñ)/(Ñ2 Ñ) α N 1 /N 2 τ α Ñ 1 /Ñ2. (4.1) Here, α and α are the homotopy classes of the inclusion induced maps defined in Proposition 2.7 and β is the unique morphism from (N 1 N)/(N 2 N) to (Ñ1 Ñ)/(Ñ2 Ñ) in the categorial Conley-Morse index I(π, S) as defined in [19] or [21]. Proposition Ñ. The definition of τ is independent of the choice of the sets N and Proof. Let N and Ñ be some other strongly π-admissible isolating neighborhoods of S with N 1 \ N 2 N and Ñ1 \ Ñ2 Ñ. First we will assume that N N and Ñ Ñ. Then we have the following diagram in K: (N 1 N )/(N 2 N ) β (Ñ1 Ñ )/(Ñ2 Ñ ) γ (N 1 N)/(N 2 N) α N 1 /N 2 β τ (Ñ1 Ñ)/(Ñ2 Ñ) γ α Ñ 1 /Ñ2. Here, γ and γ are inclusion induced maps and β is the unique morphism from (N 1 N )/(N 2 N ) to (Ñ1 Ñ )/(Ñ2 Ñ ) in the categorial Conley-Morse index I(π, S). Notice that β, γ and γ are also morphisms in I(π, S). Therefore, the upper diagram is commutative. Thus, the following diagram also commutes. (N 1 N )/(N 2 N ) β (Ñ1 Ñ )/(Ñ2 Ñ ) α γ N 1 /N 2 τ α γ Ñ 1 /Ñ2. Therefore, we have proved the proposition in the case N N and Ñ Ñ. The general case can be reduced to this particular one by considering the intersections N N and Ñ Ñ. This completes the proof. Using Proposition 4.18 we now define the set of morphisms of C(π, S) from (E, p) = (N 1 /N 2, [N 2 ]) to (Ẽ, p) = (Ñ1/Ñ2, [Ñ2]) as the singleton {τ} where τ is defined in (4.1). The morphism composition in C(π, S) is that of K. With these definitions the following result holds.

14 302 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI Proposition C(π, S) is a subcategory of K and a connected simple system. Proof. Let (E, p), (Ẽ, p) and (E, p ) be objects in C(π, S). It follows from Proposition 2.1 that there are unique FM-index pairs (N 1, N 2 ), (Ñ1, Ñ2) and (N 1, N 2) for (π, S) with N = Cl X (N 1 \ N 2 ), Ñ = Cl X (Ñ1 \ Ñ2) and N = Cl X (N 1 \ N 2) strongly π-admissible isolating neighborhoods such that (E, p) = (N 1 /N 2, [N 2 ]), (Ẽ, p) = (Ñ1/Ñ2, [Ñ2]) and (E, p ) = (N 1/N 2, [N 2]). The following diagram (N 1 N)/(N 2 N) β (Ñ1 Ñ)/(Ñ2 Ñ) α N 1 /N 2 τ α Ñ 1 /Ñ2 β (N 1 N )/N 2 N ) τ α N 1/N 2 shows that the composite of two morphisms in C(π, S) is also a morphism in C(π, S). Moreover, the commutative diagram (N 1 N)/(N 2 N) Id (N 1 N)/(N 2 N) α N 1 /N 2 Id α N 1 /N 2 shows that the identity morphism Id (E,p) of K lies in C(π, S) for every object (E, p) of C(π, S). Therefore, we have shown that C(π, S) is a subcategory of K. Since for each two objects (E, p) and (Ẽ, p) of C(π, S) there is exactly one morphism in C(π, S) from (E, p) to (Ẽ, p), we have that C(π, S) is a connected simple system. We can now make the following definition. Definition Given an isolated π-invariant set S admitting a strongly π-admissible isolating neighborhood, set H q (π, S) := Φ(C(π, S)), where Φ = H q, q Z, the q-th singular homology functor with coefficients in G (cf. (2.3)). The graded module (H q (π, S)) q Z is called the homology Conley index of S. If Φ = H q, where H q, q Z, denotes the q-th Alexander-Spanier cohomology functor (cf. (2.6)), then (H q (π, S)) q Z, where H q (π, S) := Φ(C(π, S)), q Z, is called the cohomology Conley index of S. In the remaining part of this section we will show that certain inclusion induced maps in K between objects of C(π, S) are morphisms of C(π, S). The first result is almost obvious. Proposition Let (N 1, N 2 ) and (Ñ1, Ñ2) be FM-index pairs for (π, S) with N 1 \ N 2 = Ñ1 \ Ñ2 (4.2) Sob a supervisão da CPq/ICMC

15 HOMOLOGY INDEX BRAIDS 303 such that N := Cl X (N 1 \ N 2 ) = Cl X (Ñ1 \ Ñ2) is strongly π-admissible. Then the inclusion induced map τ : N 1 /N 2 Ñ1/Ñ2 in K lies in Mor C(π,S) ((N 1 /N 2, [N 2 ]), (Ñ1/Ñ2, [Ñ2])) and so is an isomorphism in K. Proof. By (4.2) there is a commutative diagram (N 1 N)/(N 2 N) β (Ñ1 N)/(Ñ2 N) α N 1 /N 2 τ α Ñ 1 /Ñ2 (4.3) and the map β is a morphism of I(π, S). (Cf. Definition 9.2 in [21].) The proposition now follows from the definition of C(π, S). The next proposition is harder to prove. Proposition Let (N 1, N 2 ) and (Ñ1, Ñ2) be FM-index pairs for (π, S) such that Cl X (N 1 \ N 2 ) and Cl X (Ñ1 \ Ñ2) are strongly π-admissible. Assume that (N 1, N 2 ) (Ñ1, Ñ2). Then the inclusion induced map N 1 /N 2 Ñ1/Ñ2 in K lies in Mor C(π,S) ((N 1 /N 2, [N 2 ]), (Ñ1/Ñ2, [Ñ2])) and so is an isomorphism in K. The rest of this section is devoted to the proof of Proposition Let N,Y be subsets of X such that Y N. For s 0, define Y s = Y s (N) := { x X there is an s, 0 s s, such that xπs is defined, xπ [0, s ] N and xπs Y }. (4.4) Proposition Let s ]0, [ and (N 1, N 2 ) be an FM-index pair for (π, S) such that π does not explode in N 1 \ N 2. Then (N 1, N2 s (N 1)) is an FM-index pair for (π, S). Proof. We need to prove that the conditions of Definition 2.5 are satisfied for the pair (N 1, N2 s (N 1)). We only verify that N s 2 (N 1) is a closed set. (4.5) The other conditions are trivial to check. To prove (4.5) let (x n ) n be a sequence in N2 s (N 1) such that x n x as n. Since x n N2 s (N 1) for all n N, it follows that

16 304 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI for each n N, there exists an s n [0, s] such that x n π[0, s n] N 1 and x n πs n N 2. Since (s n) n is a bounded sequence, without loss of generality, we can assume that there exists an s [0, s] such that s n s as n. We need to show that x N s 2 (N 1). Now this will certainly be the case if xπs is defined. First suppose that ρ N1 (x) = 0. It follows that there exists a τ ]0, s] such that xπτ is defined and xπτ / N 1. Since x n x as n, it follows that there exists an n τ N such that x n πτ is defined and x n πτ / N 1 for all n n τ. Hence s n < τ for all n n τ. Hence s τ and so xπs is defined. Assume now that ρ N1 (x) > 0. We have two cases. First suppose that xπ [0, ρ N1 (x)[ N 1 \ N 2. Since π does not explode in N 1 \ N 2, it follows that xπρ N1 (x) is defined. Moreover, there exists a δ > 0 such that xπ(ρ N1 (x) + δ) is defined and xπ(ρ N1 (x) + δ) / N 1. Hence s n < ρ N1 (x) + δ for all n sufficiently large and so s ρ N1 (x) + δ. This implies that xπs is defined. If xπ [0, ρ N1 (x)[ N 1 \ N 2, there exists a t 0 [0, ρ N1 (x)[ such that xπt 0 N 2. If t 0 s then x N s 2 (N 1) and we are done. If t 0 > s s then xπs is defined and we are done again. This proves that N s 2 (N 1) is closed. Proposition Let S be an isolated invariant set and N be a strongly π- admissible isolating neighborhood of S. Then there is a δ 0 ]0, [ and for all δ ]0, δ 0 ], there is an isolating block B δ for S with B δ N such that 1. B δ2 B δ1, (B δ2 ) (B δ1 ) for all δ 2, δ 1 ]0, δ 0 ] with δ 2 < δ 1 ; 2. whenever (δ n ) n and (x n ) n are sequences such that δ n 0 + as n and x n B δn for all n N, then there is a subsequence of (x n ) n that converges to an element of Inv π (N). Proof. The proposition follows from the proof of Theorem I.5.1 in [21]. Lemma Let (N 1, N 2 ) and (Ñ1, Ñ2) be FM-index pairs for (π, S) such that Cl X (N 1 \ N 2 ) and Cl X (Ñ1\Ñ2) are strongly π-admissible. Then there exist an s [0, [, an isolating neighborhood L 1 of S and an index pair (L 1, L 2 ) in L 1 such that (L 1, L 2 ) (N 1, N s 2 ) and (L 1, L 2 ) (Ñ1, Ñ s 2 ), where N s 2 = N s 2 (N 1) and Ñ s 2 = Ñ s 2 (Ñ1). Proof. If S =, define L 1 = L 2 =. Let us assume that S. Define N := Cl X (N 1 \ N 2 ) Cl X (Ñ1 \ Ñ2). Thus, N is a strongly π-admissible isolating neighborhood of S. Let δ 0 ]0, [ and (B δ ) δ ]0,δ0 ] be as in Proposition We claim that there are an s 0 ]0, [ and a δ 0 ]0, δ 0 ] such that (B δ ) N2 s (N 1) Ñ2 s (Ñ1) for all s [s 0, [ and δ ]0, δ 0 ]. (4.6) Sob a supervisão da CPq/ICMC

17 HOMOLOGY INDEX BRAIDS 305 Suppose that (4.6) does not hold. Then there exist sequences (s n ) n, (δ n ) n and (x n ) n such that s n and δ n 0 + as n and for each n N, x n (B δn ) \ (N s n 2 (N 1 ) Ñ s n 2 (Ñ1)). Proposition 4.24 implies that there exists a subsequence of (x n ) n, denoted again by (x n ) n, and an x Inv π (N) such that x n x as n. It follows that x Inv π (N 1 ) Inv π (Ñ1). As x n / N sn 2 (N 1 ) Ñ sn 2 (Ñ1), it follows that for each n N, x n π [0, s n ] N 1 \ N 2 or x n π [0, s n ] Ñ1 \Ñ2. Since s n as n, this implies that x Inv + π (N 1 ) Inv + π (Ñ1) (cf. Theorem I.4.5 in [21]) and so x S. However, x n (B δn ) (B δ0 ) and this implies x S (B δ0 ) = which is a contradiction. This proves (4.6). Fix an s [s 0, [ and a δ ]0, δ 0 ]. Define L 1 := B δ and L 2 := (B δ ). Thus, L 1 is an isolating neighborhood of S and (L 1, L 2 ) is an index pair in L 1. Moreover, L 1 N and so, L 1 N 1 and L 1 Ñ1. Inclusion (4.6) implies that L 2 N2 s and L 2 Ñ 2 s. The proof is complete. Proof of Proposition Define N := Cl X (N 1 \ N 2 ) and Ñ := Cl X(Ñ1 \ Ñ2). Proposition 2.6 implies that (N 1 N, N 2 N) is an index pair in N 1 N and (Ñ1 Ñ, Ñ2 Ñ) is an index pair in Ñ1 Ñ and so (N 1 N, N 2 N) and (Ñ1 Ñ, Ñ2 Ñ) are FM-index pairs for (π, S). Lemma 4.25 implies that there are an s [0, [, an isolating neighborhood L 1 of S and an index pair (L 1, L 2 ) in L 1 such that (L 1, L 2 ) (N 1 N, (N 2 N) s ) and (L 1, L 2 ) (Ñ1 Ñ, (Ñ2 Ñ) s ), where (N 2 N) s = (N 2 N) s (N 1 N) and (Ñ2 Ñ) s = (Ñ2 Ñ) s (Ñ1 Ñ). Consider the following diagram in K: L 1 /L 2 α 1 Id (N 1 N)/(N 2 N) s α 2 (N 1 N)/(N s 2 N) α 5 L 1 /L 2 α3 (Ñ1 Ñ)/(Ñ2 Ñ) s α 4 (Ñ1 β s N 1 /N s 2 τ s Ñ)/(Ñ s 2 Ñ) α 6 Ñ 1 /Ñ s 2, where, N2 s = N2 s (N 1), Ñ2 s = Ñ 2 s (Ñ1) and the morphisms τ s and α i, i {1,..., 6}, are inclusion induced maps and β s is the unique morphism from (N 1 N)/(N2 s N) to s (Ñ1 Ñ)/(Ñ2 Ñ) lying in I(π, S). Since all the morphisms in the left rectangle lie in I(π, S), the left rectangle commutes and all the maps are isomorphisms. Hence α 6 β s = α 6 α 4 α 3 α 1 1 α 1 2. (4.7) Moreover, the full rectangle, obtained by taking out β s, is inclusion induced and commutes. Thus τ s α 5 α 2 α 1 = α 6 α 4 α 3. (4.8) Equalities 4.7 and (4.8) imply that α 6 β s = τ s α 5. In other words, the right rectangle also commutes.

18 306 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI By the definition of C(π, S) we thus see that the inclusion induced morphism τ s is a morphism in C(π, S). Now consider the following diagram of inclusion induced maps: N 1 /N 2 γ Ñ 1 /Ñ2 α α N 1 /N s 2 τ s Ñ 1 /Ñ s 2. It follows that this diagram commutes. If we show that α and α are morphisms in C(π, S), then it follows that γ is also a morphism in C(π, S) and the proposition is proved. To show that α is a morphism in C(π, S) consider the diagram (N 1 N)/(N 2 N) N 1 /N 2 (N 1 N)/(N s 2 N) N 1 /N s Since N 1 \ N s 2 N 1 \ N 2 N, it follows from the definition of C(π, S) and Proposition 4.18 that α is a morphism in C(π, S). Analogously, we prove that α is a morphism in C(π, S). The proof is complete ATTRACTOR-REPELLER PAIRS AND LONG EXACT SEQUENCES IN (CO)HOMOLOGY For the rest of this section let S be a compact π-invariant set and (A, A ) be an attractorrepeller pair of S relative to π. Let (N 1, N 2, N 3 ) be an FM-index triple for (π, S, A, A ) such that Cl X (N 1 \N 3 ) strongly π-admissible. Then, by Propositions 2.13 and 2.14, the inclusion induced sequence N 2 /N 3 i N 1 /N 3 p N 1 /N 2 of pointed spaces induces the long exact sequences H q (N 2 /N 3 ) H q(i) H q (N 1 /N 3 ) H q(p) H q (N 1 /N 2 ) q H q 1 (N 2 /N 3 ) and H q (i) H q (p) H q (N 2 /N 3 ) H q (N 1 /N 3 ) H q (N 1 /N 2 ) H q+1 (N 2 /N 3 ) By Proposition 3.17 and Definition 4.20 we obtain the long exact sequences q Sob a supervisão da CPq/ICMC

19 HOMOLOGY INDEX BRAIDS 307 H q (π, A) Hq(i) H q (π, S) Hq(p) H q (π, A ) q H q 1 (π, A) (5.1) and H q (i) H q (p) H q (π, A) H q (π, S) H q (π, A ) H q+1 (π, A). (5.2) The purpose of this section is to show that these sequences are independent of the choice of FM-index triples. More precisely we will prove the following result: Theorem If (N 1, N 2, N 3 ) and (Ñ1, Ñ2, Ñ3) are FM-index triples for (π, S, A, A ) with Cl X (N 1 \ N 3 ) and Cl X (Ñ1 \ Ñ3) strongly π-admissible, then the diagrams q H q (N 2 /N 3 ) Hq(i) H q (N 1 /N 3 ) Hq(p) H q (N 1 /N 2 ) q H q(ι A ) H q(ι S ) H q(ι A ) H q 1 (N 2 /N 3 ) H q 1(ι A ) H q (Ñ2/Ñ3) H q (Ñ1/Ñ3) H q (Ñ1/Ñ2) H q 1 (Ñ2/Ñ3) Hq (ĩ) Hq ( p) q (5.3) and H q (i) H q (p) H q (N 2 /N 3 ) H q (N 1 /N 3 ) H q (N 1 /N 2 ) H q 1 (N 2 /N 3 ) H q (ι A ) H q (ι S ) H q (ι A ) H q 1 (ι A ) H q (Ñ2/Ñ3) H q (ĩ) H q (Ñ1/Ñ3) H q ( p) H q (Ñ1/Ñ2) q q H q 1 (Ñ2/Ñ3) (5.4) commute, where ι A is the unique morphism from N 2 /N 3 to Ñ2/Ñ3 in C(π, A), ι S is the unique morphism from N 1 /N 3 to N 1 /Ñ3 in C(π, S) and ι A is the unique morphism from N 1 /N 2 to Ñ1/Ñ2 in C(π, A ). In view of Theorem 5.26 and Proposition 3.16 we have H q (i) = H q (ĩ), H q (p) = H q ( p) and q = q, q Z. Therefore the sequence 5.1 is indeed independent of the choice of an FM-index triple and exact (by Proposition 3.17). This sequence is called the homology index sequence of (π, S, A, A ). Similarly, we see that sequence (5.2) is independent of the choice of an FM-index triple and exact. This sequence is called the cohomology index sequence of (π, S, A, A ). The rest of this section is devoted to the proof of Theorem We need some preliminary results.

20 308 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI Proposition Let s ]0, [ and let (N 1, N 2, N 3 ) be an FM-index triple for (π, S, A, A ) such that π does not explode in N 1 \ N 3. Then (N 1, N 2 N3 s, N 3 s ), where N3 s = N3 s (N 1), is an FM-index triple for (π, S, A, A ). Moreover, the inclusion induced diagram of pointed spaces N 2 /N 3 (N 2 N s 3 )/N s 3 N 1 /N 3 N 1 /N s 3 N 1 /N 2 N 1 /(N 2 N s 3 ) commutes. Proof. The proof is a simple exercise, using Proposition Lemma Let (N 1, N 2, N 3 ) and (Ñ1, Ñ2, Ñ3) be FM-index triples for (π, S, A, A ) with Cl X (N 1 \ N 3 ) and Cl X (Ñ1 \ Ñ3) strongly π-admissible. Then there exist a τ [0, [ and an FM-index triple (L 1, L 2, L 3 ) for (π, S, A, A ) such that (L 1, L 2, L 3 ) (N 1, N 2 N3 τ, N3 τ ) and (L 1, L 2, L 3 ) (Ñ1, Ñ2 Ñ 3 τ, Ñ 3 τ ), where N3 τ = N3 τ (N 1 ) and Ñ 3 τ = Ñ 3 τ (Ñ1). (5.5) Proof. If S =, define L 1 = L 2 = L 3 =. Let us assume that S. Define N := Cl X (N 1 \ N 3 ) Cl X (Ñ1 \ Ñ3). Thus, N is a strongly π-admissible isolating neighborhood of S. Let δ 0 ]0, [ and (B δ ) δ ]0,δ0 ] be as in Proposition 4.24 (with the present choice of the set N). Proceeding as the proof of Lemma 4.25 we see that there are an s 0 ]0, [ and a δ 0 ]0, δ 0 ] such that (B δ ) N3 s (N 1) Ñ 3 s (Ñ1) for all s [s 0, [ and δ ]0, δ 0 ]. (5.6) Fix s 1 [s 0, [ and define N 1 := N 1, N 2 := N 2 N s1 3 (N 1 ), N 3 := N s1 3 (N 1 ), Ñ 1 := Ñ1, Ñ 2 := Ñ2 Ñ s 1 3 (Ñ1) and Ñ 3 := Ñ s 1 3 (Ñ1). It follows that from Proposition 5.27 that (N 1, N 2, N 3) and (Ñ 1, Ñ 2, Ñ 3) are FM-index triples for (π, S, A, A ). Moreover, We claim that (B δ ) N 3 Ñ 3 for all δ ]0, δ 0 ]. (5.7) there exists an s 0 ]0, [ and a δ 1 ]0, δ 0 ] such that B δ (N 3 Ñ 3) N 3 s (N 1) Ñ 3 s ( Ñ 1) for all s [s 0, [ and δ ]0, δ 1 ]. (5.8) Suppose that (5.8) does not hold. Then there exist sequences (s n ) n, (δ n ) n and (x n ) n such that s n and δ n 0 + as n and for each n N, x n (B δn (N 3 Ñ 3 )) \ (N 3 s n (N 1) Ñ 3 s n (Ñ 1)). Sob a supervisão da CPq/ICMC

21 HOMOLOGY INDEX BRAIDS 309 Proposition 4.24 implies that there exists a subsequence of (x n ) n, denoted again by (x n ) n, and an x Inv π (N) such that x n x. Hence x Inv π (Cl X (N 1 \ N 3 )) Inv π (Cl X (Ñ1 \ Ñ3)). Since x n / N 3 s n (N 1) Ñ 3 s n (Ñ 1) for all n N, it follows that for each n N, x n π [0, s n ] N 1 \ N 3 or x n π [0, s n ] Ñ 1 \ Ñ 3. Since s n as n, this implies that x Inv + π (Cl X (N 1 \ N 3)) Inv + π (Cl X (Ñ 1 \ Ñ 3)) (cf. Theorem I.4.5 in [21]). Since N 1 = N 1, N 3 N 3, Ñ 1 = Ñ1 and Ñ3 Ñ 3, it follows that x Inv + π (Cl X (N 1 \ N 3 )) Inv + π (Cl X (Ñ1 \ Ñ3)) and so x S. We thus obtain that x S (N 3 Ñ 3) = which is a contradiction and so our claim is proved. Fix an s [s 0, [ and a δ ]0, δ 1 ]. Define L 1 := B δ, L 2 := (B δ (N 2 Ñ 2)) (B δ (N 3 Ñ 3)) and L 3 := B δ (N 3 Ñ 3). Since B δ N N 1 Ñ1 we obtain that L 1 N 1 and L 1 Ñ1. (5.9) Inclusion (5.8) implies that L 2 N 2 N 3 s (N 1), L 2 Ñ 2 Ñ 3 s ( Ñ 1), L 3 N 3 s (N 1) and L 3 N 3 s (Ñ 1). Let x N 3 s (N 1). Thus, there is an s [0, s] such that xπs is defined, xπ [0, s ] N 1 = N 1 and xπs N 3. Since xπs N 3 = N s 1 3 (N 1 ), it follows that there exists an s [0, s 1 ] such that (xπs )πs is defined, (xπs )π [0, s ] N 1 and (xπs )πs N 3. Thus, xπ [0, s + s ] N 1 and xπ(s + s ) N 3 with 0 s + s s + s 1. (s+s In other words, x N 1) 3 (N 1 ). Therefore, where τ := s + s 1. Moreover and We similarly obtain that L 2 N 2 N 3 s 1 (N 1 ) N 3 (s+s 1 ) (N 1 ) N 2 N 3 τ (N 1 ), (5.10) L 3 N 3 s (N 1 ) N 3 (s+s 1 ) (N 1 ) = N 3 τ (N 1 ). (5.11) L 2 Ñ2 Ñ s 1 3 (Ñ1) Ñ (s+s1) 3 (Ñ1) Ñ2 Ñ τ 3 (Ñ1) (5.12) L 3 Ñ 3 s ( Ñ 1) Ñ (s+s1) 3 (Ñ1) = Ñ 3 τ (Ñ1). (5.13) Inclusions (5.9), (5.10), (5.12), (5.11) and (5.13) imply the inclusions in (5.5). To finish the proof we need to show that (L 1, L 2, L 3 ) is an FM-index triple for (π, S, A, A ). We claim that (L 1, L 3 ) is an FM-index pair for (π, S). Indeed, notice that S Int X (L 1 ) and S (N 3 Ñ 3) =. Thus, S Int X (L 1 \ L 3 ) and so S Inv π (Cl X (L 1 \ L 3 )). On the

22 310 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI other hand, L 1 \ L 3 N 1 \ N 3 and this implies that Inv π (Cl X (L 1 \ L 3 )) Inv π (Cl X (N 1 \ N 3)) = S. Let x L 3 and t 0 be such that xπ [0, t] L 1. Hence, x L 1 (N 3 Ñ 3) and so x N 3 Ñ 3. Since N 3 is N 1-positively invariant and Ñ 3 is Ñ 1-positively invariant, it follows that xπ [0, t] N 3 Ñ 3 so xπ [0, t] L 3 and so L 3 is L 1 -positively invariant. Let x L 1 be such that xπt / L 1 for some t > 0. Since (B δ, B δ ) is an index pair in B δ, there is a t [0, t[ such that xπ [0, t ] L 1 and xπt (B δ ). Inclusion (5.7) implies that xπt N 3 Ñ 3 and so xπt L 3. Thus L 3 is an exit ramp for L 1. The proof of our claim is complete. We now claim that (L 2, L 3 ) is an FM-index pair for (π, A). Note that A Int X (N 2) Int X (Ñ 2), A S Int X (B δ ) and A (B δ (N 3 Ñ 3)) = so A Int X (L 2 \ L 3 ) and so A Inv π (Cl X (L 2 \ L 3 )). On the other hand, L 2 \ L 3 N 2 \ N 3 and this implies that Inv π (Cl X (L 2 \ L 3 )) Inv π (Cl X (N 2 \ N 3)) = A. Let x L 3 and t 0 be such that xπ [0, t] L 2. Thus x N 3 Ñ 3. Since N 3 is N 2-positively invariant and Ñ 3 is Ñ 2-positively invariant, it follows that xπ [0, t] N 3 Ñ 3. Recall that L 2 B δ. Thus, xπ [0, t] L 3 and so L 3 is L 2 -positively invariant. Let x L 2 be such that xπt / L 2 for some t > 0. We need to show that there exists a t [0, t[ such that xπ [0, t ] L 2 and xπt L 3. Since xπt / L 2, it follows that xπt / B δ (N 2 Ñ 2) and xπt / B δ (N 3 Ñ 3). Suppose first x B δ (N 2 Ñ 2). Set t := ρ Bδ (N 2 Ñ 2 )(x). By (2.1), xπr is defined and xπr B δ (N 2 Ñ 2) for all r [0, t [. Therefore, we cannot have t < t so t [0, t]. Moreover, xπ[0, t ] B δ (N 2 Ñ 2), since B δ (N 2 Ñ 2) is closed. By (2.2) we have that t = ρ Bδ (x) or t = ρ N 2 (x) or t = ρñ (x). In the first case it follows that xπt B δ 2 B δ (N 3 Ñ 3) = L 3 ; in the second case xπt N 3 so xπt B δ (N 3 Ñ 3) = L 3 and in the third case xπt Ñ 3 so xπt B δ (N 3 Ñ 3) = L 3. Suppose now that x B δ (N 3 Ñ 3). In this case, define t := 0. The proof of the lemma is complete. Proof of Theorem Let (N 1, N 2, N 3 ) and (Ñ1, Ñ2, Ñ3) be two FM-index triples for (π, S, A, A ) with Cl X (N 1 \ N 3 ) and Cl X (Ñ1 \ Ñ3) strongly π-admissible. Let s > 0 and (L 1, L 2, L 3 ) be an FM-index triple for (π, S, A, A ) such that the conclusions of Lemma 5.28 holds. Proposition 4.22 and Proposition 5.27 imply that the inclusion induced diagram (5.14) of pointed spaces commutes. Passing to homology in diagram (5.14) we obtain the commutative diagram (5.15) (in which we set M 2 := N 2 N3 s and M 2 := Ñ 2 Ñ 3 s ) made of four long homology ladders. An application of Proposition 4.22 shows that the vertical morphisms in diagram (5.15) are isomorphisms. Thus we can reverse the vertical arrows in the second and fourth ladders. Composing the resulting ladders, we obtain the commutative diagram (5.3), completing the proof in the singular homology case. Sob a supervisão da CPq/ICMC

23 HOMOLOGY INDEX BRAIDS 311 i N 2 /N 3 N 1 /N 3 (N 2 N s 3 )/N s 3 N 1 /N s 3 p N 1 /N 2 N 1 /(N 2 N s 3 ) (5.14) L 2 /L 3 (Ñ2 Ñ s s )/Ñ 3 3 L 1 /L 3 Ñ 1 /Ñ s 3 L 1 /L 2 Ñ 1 /(Ñ2 Ñ s 3 ) Ñ 2 /Ñ3 ĩ Ñ 1 /Ñ3 p Ñ 1 /Ñ2 H q (N 2 /N 3 ) Hq(i) H q (N 1 /N 3 ) Hq(p) q H q (N 1 /N 2 ) H q 1 (N 2 /N 3 ) H q (M 2 /N3 s ) H q (N 1 /N3 s ) H q (N 1 /M 2 ) H q 1 (M 2 /N3 s ) (5.15) H q (L 2 /L 3 ) H q (L 1 /L 3 ) H q (L 1 /L 2 ) H q 1 (L 2 /L 3 ) H q ( M 2 /Ñ 3 s ) H q (Ñ1/Ñ 3 s ) H q (Ñ1/ M 2 ) H q 1 ( M 2 /Ñ 3 s ) H q (Ñ2/Ñ3) Hq (ĩ) H q (Ñ1/Ñ3) Hq ( p) H q (Ñ1/Ñ2) q H q 1 (Ñ2/Ñ3) The proof for the Alexander-Spanier cohomology is analogous. 6. MORSE DECOMPOSITIONS AND (CO)HOMOLOGY INDEX BRAIDS Recall that a strict partial order on a set P is a relation P P which is irreflexive and transitive. As usual, we write x y instead of (x, y). The symbol < will be reserved for the less-than-relation on R. For the rest of this paper, unless specified otherwise, let P be a fixed finite set and be a fixed strict partial order on P. A set I P is called a -interval if whenever i, j, k P, i, k I and i j k, then j I. By I( ) we denote the set of all -intervals in P. A set I is called a -attracting interval if whenever i, j P, j I and i j, then i I. By A( ) we denote the set of all -attracting intervals in P. Of course, A( ) I( ). An adjacent n-tuple of -intervals is a sequence (I j ) n j=1 of pairwise disjoint -intervals whose union is a -interval and such that, whenever j < k, p I j and p I k, then p p

24 312 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI (i.e. p p or else p and p are not related by ). By I n ( ) we denote the set of all adjacent n-tuples of -intervals. Let S X be a compact π-invariant set. A family (M i ) i P of subsets of S is called a -ordered Morse decomposition of S if the following properties hold: 1. The sets M i, i P, are closed, π-invariant and pairwise disjoint. 2. For every full solution σ of π lying in S either σ(r) M k for some k P or else there are k, l P with k l, α(σ) M l and ω(σ) M k. Let S be a compact invariant set and (M i ) i P be a -ordered Morse decomposition of S. If A, B X then the (π, S)-connection set CS π,s (A, B) from A to B is the set of all points x X for which there is a solution σ : R S of π with σ(0) = x, α(σ) A and ω(σ) B. For an arbitrary -interval I set M(I) = M π,s (I) = (i,j) I I CS π,s (M i, M j ). An index filtration for (π, S, (M p ) p P ) is a family N = (N(I)) I A( ) of closed subsets of X such that 1. for each I A( ), the pair (N(I), N( )) is an FM-index pair for (π, M(I)), 2. for each I 1, I 2 A( ), N(I 1 I 2 ) = N(I 1 ) N(I 2 ) and N(I 1 I 2 ) = N(I 1 ) N(I 2 ). N is called strongly π-admissible if N(P ) is strongly π-admissible. An existence result for such index filtrations was established in [12]. Let N be a strongly π-admissible index filtration for (π, S, (M p ) p P ). For J I( ) the set M(J) is an isolated invariant set and we write H q (J) = H q (π, J) := H q (π, M(J)), q Z. If (I, J) I 2 ( ), then (M(I), M(J)) is an attractor-repeller pair in M(IJ), where IJ := I J. Let B be the set of all p P \ (IJ) for which there is a p IJ with p p. It follows that B, BI, BIJ A( ). Moreover, (N(BIJ), N(BI), N(B)) is an FM-index triple for (π, M(IJ), M(I), M(J)) with Cl X (N(BIJ) \ N(B)) strongly π-admissible. The inclusion induced sequence N(BI)/N(B) i I,J N(BIJ)/N(B) p I,J N(BIJ)/N(BI) induces the homology index sequence H q (I) H q(i I,J ) H q (IJ) H q(p I,J ) H q (J) I,J,q H q 1 (I) of (π, M(IJ), M(I), M(J)). Let (I, J, K) I 3 ( ) and define H := {p P there is a p IJK with p p }. It follows that H A( ) and (H, I, J, K) I 4 ( ). Hence, Sob a supervisão da CPq/ICMC

25 HOMOLOGY INDEX BRAIDS 313 HI, HIJ, HIJK A( ). Define N 1 := N(HIJK), N 2 := N(HIJ), N 3 := N(HI) and N 4 := N(H). We obtain the following inclusion induced diagram N 3 /N 4 i 1 (6.1) i 4 i 2 N 1 /N 4 p 4 p 3 N 1 /N 2 p 2 N 2 /N 4 p 1 i 3 N 1 /N 3 N 2 /N 3 of pointed spaces. Applying Propositions 2.11 and 2.12 to diagram (6.1) and then using the -operation together with Proposition 3.17 we obtain the commutative diagram H q(i 4) H q (p 2 ) H q (I) H q (i 1 H ) q+1 (K) 2,q H q (IJ) H q (i 2 ) H q (p 1 ) H q (IJK) Hq(p 4) H q (J) H q(i 3) H q (JK) H q (p 3 ) 4,q H q (K) 2,q H q 1 (I) H q 1 (i 1 ) H q 1 (IJ) H q 1(p 1) Hq 1(i 2) H q 1 (J) H q 1 (IJK) 3,q 3,q 1,q H q 1 (i 4 ) (6.2) Since all morphisms in diagram (6.2) are in the long exact homology sequences of the appropriate attractor-repeller pairs, it follows that diagram (6.2) is independent of the choice of an admissible index filtration for(π, S, (M p ) p P ). The following concept is thus well defined. Definition ([9], [12]) The collection of all the homology indices H q (π, M(J)), q Z, J I( ), and all the maps H q (i I,J ), H q (p I,J ) and I,J,q, (I, J) I 2 ( ) is called the homology index braid of (π, S, (M p ) p P ). We denote it by H(π, S, (M p ) p P ).

26 314 MARIA C. CARBINATTO AND KRZYSZTOF P. RYBAKOWSKI For the rest of this section assume that, for i = 1, 2, π i is a local semiflow on the metric space X i, S i is an isolated invariant set and (M p,i ) p P is a -ordered Morse decomposition of S i, relative to π i. Write M i (I) = M πi,s i (I), H i (I) = H(π i, M i (I)) and H i := H(π i, S i, (M p,i ) p P ), for i = 1, 2 and I I( ). Suppose θ := (θ(j)) J I( ) is a family θ(j): H 1 (J) H 2 (J), J I( ), of Γ-module homomorphisms such that, for all (I, J) I 2 ( ), the diagram H 1,q (I) H q(i I,J ) H 1,q (IJ) H q(p I,J ) H 1,q (J) I,J H 1,q 1 (I) (6.3) θ q(i) θ q(ij) H 2,q (I) H 2,q (IJ) Hq (i I,J ) Hq (p I,J ) θ q(j) H 2,q (J) I,J θ q 1(I) H 2,q 1 (I) commutes. Then we say that θ is a morphism from H 1 to H 2 and we write θ : H 1 H 2. If each θ(j) is an isomorphism, then we say that θ is an isomorphism and that H 1 and H 2 are isomorphic homology index braids. Remark If H 1 and H 2 are isomorphic homology index braids, then, by Proposition 1.5 in [10], H 1 and H 2 determine the same collection of connection matrices and the same collection of C-connection matrices. We will now introduce an important class of morphisms between homology index braids. Let N i = (N i (I)) I A( ) be a strongly π i -admissible index filtration for (π i, S i, (M p,i ) p P ), i = 1, 2. Assume the nesting property N 1 (I) N 2 (I), I A( ). For J I( ) choose I, K A( ) with (I, J) I 2 ( ) and K = IJ. Then, for i = 1, 2, (N i (K), N i (I)) is an FM-index pair for M i (J), relative to π i. The inclusion induced map α: N 1 (K)/N 1 (I) N 2 (K)/N 2 (I) induces a homomorphism defined by θ(j) = θ N1,N 2 (J): H(π 1, M 1 (J)) H(π 2, M 2 (J)) θ q (J) := H q (α), q Z. Of course, this homomorphism depends on the choice of N i, i = 1, 2, but we claim that θ q (J), q Z, is independent of the choice of I and K. (6.4) In fact, if I and K A( ) are such that (I, J) I 2 ( ) and K = I J then property (2) of index filtrations implies that N i (K)\N i (I) = N i (K )\N i (I ), i = 1, 2, (see Proposition 3.5 in [9] and its proof, which is also valid in our case) so there is an inclusion induced, commutative, diagram of pointed spaces Sob a supervisão da CPq/ICMC